Category Archives: proof

Totient sums

Posted on October 7, 2016 by

I took a bit of a break to travel to Japan for a conference, but I’m back now to continue the series I started with Post Without Words #10, a follow-up post, and Post Without Words #11. Recall that we … Continue reading →

Posted in geometry, pattern, pictures, posts without words, proof | Tagged , , , , , | 4 Comments

Post without words #11

Posted on September 8, 2016 by
Posted in geometry, pattern, pictures, posts without words, proof | Tagged , , , , , | 7 Comments

The route puzzle

Posted on June 17, 2016 by

While poking around some old files I came across this puzzle: (Click for a larger version.) I didn’t make it, and I have no idea where I got it from (do you know?). But in any case, wherever it comes … Continue reading →

Posted in arithmetic, challenges, number theory, proof, puzzles | Tagged , , , , , , | 7 Comments

Golden numbers are Fibonacci

Posted on March 19, 2016 by

This post is fourth in a series, proving the curious fact that is a Fibonacci number if and only if one (or both) of or is a perfect square; we call numbers of this form golden numbers. Last time, I … Continue reading →

Posted in arithmetic, computation, famous numbers, fibonacci, proof | Tagged , , , , , | 2 Comments

Fibonacci numbers are golden

Posted on February 28, 2016 by

Recall that a “golden number” (this is not standard terminology) is a number such that one (or both) of or is a perfect square. In this post, I’ll explain Gessel’s proof that every Fibonacci number is golden. First, we need … Continue reading →

Posted in arithmetic, computation, famous numbers, fibonacci, proof | Tagged , , , , , | 1 Comment

Testing Fibonacci numbers: the proofs

Posted on February 27, 2016 by

In my last post I stated this surprising theorem: is a Fibonacci number if and only if one of is a perfect square. If one of is a perfect square, then let’s say that is a “golden number” (a nod, … Continue reading →

Posted in arithmetic, computation, famous numbers, fibonacci, proof | Tagged , , , | 1 Comment

The chocolate bar game: losing positions proved

Posted on January 23, 2016 by

In my last post I claimed that the losing positions for the chocolate bar game are precisely those of the form (or the reverse), that is, in binary, positions where one coordinate is the same as the other with any … Continue reading →

Posted in games, pattern, pictures, proof, solutions | Tagged , , , , , , , , | 4 Comments